Some aspects of simultaneous rational approximation

Simultaneous Rational Approximation to Binomial Functions

We apply Padé approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel’dman and Rickert, proving, for example, that max {∣∣∣√2− p1/q∣∣∣ , ∣∣∣√3− p2/q∣∣∣} > q−1.79155 for q > q0 (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be d...

On Some Constructive Method of Rational Approximation

We study a constructive method of rational approximation of analytic functions based on ideas of the theory of Hankel operators. Some properties of the corresponding Hankel operator are investigated. We also consider questions related to the convergence of rational approximants. Analogues of Montessus de Ballore’s and Gonchars’s theorems on the convergence of rows of Padé approximants are proved.

Some New Aspects of Rational Interpolation

A new algorithm for rational interpolation based on the barycentric formula is developed; the barycentric representation of the rational interpolation function possesses various advantages in comparison with other representations such as continued fractions: it provides, e.g., information concerning the existence and location of poles of the interpolant.

Best Simultaneous Approximation on Small Regions by Rational Functions

We study the behavior of best simultaneous (lq , Lp)-approximation by rational functions on an interval, when the measure tends to zero. In addition, we consider the case of polynomial approximation on a finite union of intervals. We also get an interpolation result.

On the Furtwangler algorithm for simultaneous rational approximation